AbstractFor some years I have been engaged in a close reading of early
Italian abbacus books and related material from the Ibero-Provençal orbit and in
comparison of this material with Arabic mathematical writings. At the 7th North
African Meeting on the History of Arab Mathemat­ics in Marrakesh in 2002 I
presented the first outcome of this investigation: namely that early Italian
abbacus algebra was not influenced by the Latin algebraic writings of the
12th-13th centuries (neither the translations of al-Khwārizmī nor the works
of Fibonacci); instead, it received indirect inspiration from a so far unknown
link to the Arabic world, viz to a level of Arabic algebra (probably linked to
mu‛āmalāt
mathematics) of which very little is known. At the 8th Meeting in Tunis in 2004
I presented a list of linguistic clues which, if applied to Arabic material,
might enable us to say more about the links between the Romance abbacus
tradition and Arabic mu‛āmalāt
teaching.

Here I investigate a number of problem types and
techniques which turn up in some but not necessarily in all of the following
source types:

– Romance abbacus writings,

– Byzantine writings of abbacus type,

– Arabic mathematical writings of various kinds,

– Sanskrit mathematical writings,

in order to display the intricacies of the links
between these – intricacies which force us to become aware of the shortcomings
of our current knowledge, and hence formulate questions that go beyond the
answers I shall be able to present.

Institute for the Study of the Ancient World,
New York University, November 12–13, 2010

Abstract

Those who nowadays work on the
history of advanced-level Babylonian mathematics do so as if everything had
begun with the publication of Neugebauer’s Mathematische Keilschrift-Texte
from 1935–37 and Thureau-Dangin’s Textes mathématiques babyloniens from
1938, or at most with the articles published by Neugebauer and Thureau-Dangin
during the few preceding years. Of course they/we know better, but often that is
only in principle. The present paper is a sketch of how knowledge of Babylonian
mathematics developed from the beginnings of Assyriology until the 1930s, and
raises the question why an outsider was able to create a breakthrough where
Assyriologists, in spite of the best will, had been blocked. One may see it as
the anatomy of a particular “Kuhnian revolution”.

Fibonacci during his boyhood
went to Bejaïa, learned about the Hindu-Arabic numerals there, and continued to
collect information about their use during travels to the Arabic world. He then
wrote the Liber abbaci, which with half a century’s delay inspired the creation
of Italian abbacus mathematics, later adopted in Catalonia, Provence, Germany
etc.
This story is well known – too well known to be true, indeed.
There is no
doubt, of course, that Fibonacci learned about Arabic (and Byzantine) commercial
arithmetic, and that he presented it in his book. He is thus a witness (with a
degree of reliability which has to be determined) of the commercial mathematics
thriving in the commercially developed parts of the Mediterranean world.
However, much evidence – presented both in his own book, in later Italian
abbacus books and in similar writings from the Iberian and the Provençal regions
– shows that the Liber abbaci did not play a central role in the later adoption.
Romance abbacus culture came about in a broad process of interaction with Arabic
non-scholarly traditions, interaction at first apparently concentrated in the
Iberian region.

Institute for the History of the Natural Sciences, Chinese Academy of Science,
1-2 September 2011◄

Abstract

In 1942, Edgar Zilsel proposed
that the sixteenth-seventeenth-century emergence
of Modern science was produced neither by the university tradition, nor by the
Humanist current of Renaissance culture, nor by craftsmen or other practitioners
but through an interaction between all three in which all were indispensable
for the outcome. He only included mathematics via its relation to the
'quantitative spirit'. The present study tried to apply Zilsel's perspective to
the emergence
of the Modern algebra of Viète and Descartes (etc.), by tracing the reception of
algebra within the Latin-Universitarian tradition, the Italian abbacus tradition
and Humanism, and the exchanges between them, from the twelfth through the
late sixteenth and early seventeenth century.

The fourth-millennium state formation process in Mesopotamia was intimately
linked to accounting and to a writing system created exclusively as support for
accounting. This triple link between the state, mathematics and the scribal
craft lasted until the end of the third millennium, whereas the connection
between learned scribehood and accounting mathematics lasted another four
hundred years. Though practical mathematics was certainly not unknown in the
Greco-Hellenistic-Roman world, a similar integration was never realized.
Social prestige usually goes together with utility for the
power structure (not to be confounded with that mere utility for those in power
which characterizes a working and tax/tribute-paying population), and until the
1600 BCE scribes appears to have enjoyed high social prestige.
From the moment writing and accounting was no longer one
activity among others of the ruling elite (c. 2600 BCE) but the task of a
separate profession, this profession started exploring the capacity of the two
professional tools, writing and calculation.Within the field of mathematics,
this resulted in the appearance of “supra-utilitarian mathematics”: mathematics
which to a superficial inspection appears to deal with practical situations but
which, without having theoretical pretensions, goes beyond anything which could
ever be encountered in real practice. After a setback in the late third
millennium, supra-utilitarian mathematics reached a high point – in particular
in the so-called “algebra” during the second half (1800–1600) of the “Old
Babylonian” period.
Analysis of the character and scope of this “algebraic”
discipline not only highlights the difference between theoretical and high-level
supra-utilitarian mathematics, it also makes some features of Greek theoretical
mathematics stand out more clearly. Babylonian “algebra” was believed by
Neugebauer (and by many after him on his authority) have inspired Greek
so-called “geometric algebra”. This story, though not wholly mistaken, is today
in need of reformulation; this reformulation throws light on one of the
processes that resulted in the creation of Greek theoretical mathematics.

Writing, as well as various mathematical
techniques, were created in proto-literate Uruk in order to serve accounting,
and Mesopotamian mathematics as we know it was always expressed in writing. In
so far, mathematics generically regarded was always part of the
generic written tradition.

However, once we move away from the generic
perspective, things become much less easy. If we look at elementary numeracy
from Uruk IV until Ur III, it is possible to point to continuity and thus
to a “tradition”, and also if we look at place-value practical computation from
Ur III onward – but already the relation of the latter tradition to type of
writing after the Old Babylonian period is not well elucidated by the sources.

Much worse, however, is the
situation if we consider the sophisticated mathematics created during the Old
Babylonian period. Its connection to the school institution and the new literate
style of the period is indubitable; but we find no continuation similar to that
descending from Old Babylonian beginnings in fields like medicine and extispicy.
Still worse, if we look closer at the Old Babylonian material, we seem to be
confronted with a small swarm of attempts to create traditions, but all
rather short-lived. The few mathematical texts from the Late Babylonian
(including the Seleucid) period also seem to illustrate attempts to create
traditions rather than to be witnesses of a survival for sufficiently long to
deserve the label “traditions”.